If the "serious crime" is selling counterfeit goods, which are advertised as such, then I think it's overreach. As it is, ISPs and hosting companies are really good at detecting and shutting sites with malicious content and ones that are running bot-nets and such. Additionally, they seem to have all the power they need to take down child porn sites, so what "serious crimes" exactly are we talking about?
Somehow I doubt we're talking about fake *** boots or the like. SOCA deal with things like drug/child/human trafficking, illegal firearms, that sort of thing. The US equivalent is probably the FBI.
I shouldn't have said serious crime as I mean't a class above that. I don't think that a counterfit goods website should be shut down for example without due process. Its not that serious crime websites should be shut down its that the judicial review process is not required before the action, like arresting someone without an arrest warrant. The only ones that should be shut down quickly in the manner suggested are those beyond a serious crime. The issue is oversite more than an issue with the power to do it.
Although your statement is wrong it is right in some ways. So many people use google that to an extent if google doesn't have your site listed then it may as well be invisible.
I feel this quote applies: These aren't exactly things that people just do a search on Google for, not if they don't want to be promptly arrest at least. For a less serious crime you're probably on to something. It'd be like the advertised sites you see at the top, except in reverse.